Assume \(\lambda > \lambda _c(\phi )\) .
Let \({\varepsilon }_1 \in (0,{\varepsilon }/3)\)  be chosen such that if \((X_x)_{x \in \mathbb {Z}^2}\)  is
a 7-dependent Bernoulli random
field on \(\mathbb {Z}^2\)  with \({\mathbb {P}}[X_x=1] > 1- 5{\varepsilon }_1\)  for all \(x \in \mathbb {Z}^d\) ,
than for all \(s >0\)  and all \(x \in \mathbb {Z}^2 \cap B(s)\) , there
is a lattice path of 1's from \(x \)  to \({\bf o}\)  in \(\mathbb {Z}^2 \cap B(s)\)  with
probability greater than \(1- {\varepsilon }/3\) .
The proof that such an \({\varepsilon }_1\)  exists is standard, using
e.g. [1]} and a Peierls argument (e.g. [2]}).
